Following [1]}, we will first construct a quotient of the action of the graded unipotent group \(\widehat{U}\) , and then use reductive GIT to construct the residual quotient by \(\overline{L}\) . In \(§\)REF  and \(§\)REF , we explain the central results in [1]}, [3]}, which give the construction of a quasi-projective geometric quotient of the \(H\) -action on an open stable set of \(Y\)  admitting a Hilbert–Mumford type description together with a natural projective completion. The first step is to show that one can construct a projective quotient of the \(H\) -action if one assumes the stabilisers are reasonably well-behaved (see Definition REF  below, which one can think of as `semistability coinciding with stability' condition); we describe this case in \(§\)REF  below (cf. Theorems REF  and REF ). To prove the more general statement without assuming this condition on the stabilisers, one performs a sequence of equivariant blow-ups to arrange for this condition to hold on the resulting blow-up and then we construct a quotient of an open set in \(Y\)  as an open subset of the projective quotient of the blow-up; this is outlined in \(§\)REF .
